Difference between revisions of "Activities-Real numbers"
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'''Approaches to solutions'''<br> | '''Approaches to solutions'''<br> | ||
==The concepts that a pupil must know are== | ==The concepts that a pupil must know are== | ||
+ | #What is an integer? | ||
+ | #What is an even integer? | ||
+ | #What is an odd integer? | ||
+ | #X and ( x+1) are consecutive integers and x(x+1) is the representation of the product | ||
+ | #Pupil should have the concept of distributive property of integers | ||
+ | #The pupil must have an opportunity for an arguement that the proof is true even of negative integers | ||
+ | #Pupil must have a sound understanding of Euclid's lemma | ||
+ | #The difference between mathematical proofs and Verification/Justification-the scope of mathematical proofs is beyond verification-Higher order skill in problem solving | ||
+ | ==Method -1: Solution by cases== | ||
+ | Proposition: x is an integer. i.e., x=m, where m is an integer<br> | ||
+ | Conclusion : x(x+1) is an even integer. i.e., x(x+1) =2K<br> | ||
+ | |||
+ | Pupil can solve this in several ways viz., proofs by cases. | ||
+ | ===Case -1: x is an even integer === | ||
+ | Then x=2k+0 = 2K, <math>K in Z </math> by Euclid's lemma<br> | ||
+ | |||
+ | Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2<br> | ||
+ | Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2 | ||
+ | '''Abstract thinking:'''<br> | ||
+ | b) a (q<br> | ||
+ | -bq<br> | ||
+ | ......... <br> | ||
+ | (a - bq)<br> | ||
+ | ......... <br> | ||
+ | |||
+ | [[Category:Types of Numbers]] |
Latest revision as of 11:03, 7 November 2019
Problem 1
- For every integer prove that x(x+1) is an even integer (Problem related to mathematical proofs in Chapter 1)
Approaches to solutions
The concepts that a pupil must know are
- What is an integer?
- What is an even integer?
- What is an odd integer?
- X and ( x+1) are consecutive integers and x(x+1) is the representation of the product
- Pupil should have the concept of distributive property of integers
- The pupil must have an opportunity for an arguement that the proof is true even of negative integers
- Pupil must have a sound understanding of Euclid's lemma
- The difference between mathematical proofs and Verification/Justification-the scope of mathematical proofs is beyond verification-Higher order skill in problem solving
Method -1: Solution by cases
Proposition: x is an integer. i.e., x=m, where m is an integer
Conclusion : x(x+1) is an even integer. i.e., x(x+1) =2K
Pupil can solve this in several ways viz., proofs by cases.
Case -1: x is an even integer
Then x=2k+0 = 2K, by Euclid's lemma
Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2
Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2
Abstract thinking:
b) a (q
-bq
.........
(a - bq)
.........